Suppose your friend is walking down University Avenue at a speed of $2$ blocks per minute. You want to know where she will be in five minutes so that you can surprise her. This situation can be modeled by the pure-time differential equation $\frac{d x}{d t} = 2$. The state variable $x$, is a function of time so is sometimes written as $x(t)$. The value $x(t)$ represents the number of blocks your friend is down University Avenue from Central Avenue , where $t$ represents time in minutes. Finding the solution to the differential equation $\frac{d x}{d t}=f(t)$ means explicitly finding a function $x(t)$ whose derivative is equal to $f(t)$.
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How can we find $x$ as a function of $t$? There are several approaches to solving pure-time differential equations, and one is “guess and check.” Use your intuition to guess a solution to $\frac{d x}{d t} = 2$:
$x{\left (t \right )}=$
Now verify that this is a solution by taking the derivative: $\frac{d x}{d t}=$
Hint
Can you come up with some function of $t$ that is $2$ after you differentiate it?
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Is the solution you found in part a the only solution to the differential equation? To help you find another function that has the same derivative, consider: what is the derivative of a constant?
What happens to the derivative of a function if you add a constant to the function?
Find a different solution to the differential equation: $x{\left (t \right )} = $
.
If we want to represent all possible solutions to this differential equation, we have to add an arbitrary constant. Let's call this constant $C$. Then $x{\left (t \right )}=$
is a solution for any choice of $C$. This is called the general solution.
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If we can add an arbitrary constant to our solution to get another solution, how many solutions are there to the differential equation $\frac{d x}{d t} = 2$?
Only one of these solutions will actually tell you where your friend is at any given time, though. How can you decide which one?
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We can't figure out where your friend will be in five minutes without knowing where she is at some point in time. Suppose that right now ($t=0$), she is already $4$ blocks from Central Ave; in other words, we suppose we have the initial condition $x(0)=4$. What is the value of $C$ such that $x{\left (0 \right )} = 4$? To determine $C$, first plug in $t=0$ into the general solution from part b to determine that $x(0)=$
. Therefore, to match this condition, we need to set $C=$
.
The particular solution that describes your friend's location with the initial condition $x{\left (0 \right )} = 4$ is $x{\left (t \right )} = $
. How many blocks from Central Ave will your friend be in five minutes? $x(5)=$
.
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What if the previous information about your friend was incorrect and, instead, the correct initial condition is that your friend is $7$ blocks from Central Ave at $t=0$? The new function describing her location is $x{\left (t \right )}=$
. In five minutes, she will be
blocks from Central Ave.
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Use the applet to plot the two functions from parts d and e describing your friend's position.
Feedback from applet
Hint
To draw the solutions using the applet, simply drag the points so that lines represent the two particular solutions you find. It doesn't matter which line you use for which solution.
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